A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of (a) the fundamental?

Verified step by step guidance

1. First, understand that the fundamental frequency of a stretched string is the lowest frequency at which the string can vibrate. It corresponds to the string vibrating in one segment, with nodes at each end.
2. The speed of a wave on a string is given by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi> = <mi>f</mi><mi>&#x3bb;</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> is the wave speed, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is the frequency, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3bb;</mi></math> is the wavelength.
3. For the fundamental frequency, the wavelength is twice the length of the rope because the rope vibrates in one segment. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3bb;</mi> = 2L</math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the length of the rope.
4. Substitute the given length of the rope into the wavelength formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3bb;</mi> = 2 imes 1.50 ext{ m}</math>.
5. To find the frequency of the fundamental, rearrange the wave speed formula to solve for frequency: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi> = <mfrac><mi>v</mi><mi>&#x3bb;</mi></mfrac></math>. Substitute the known values of wave speed and wavelength to find the frequency.

Key Concepts

Wave Speed

Wave speed is the rate at which a wave propagates through a medium, calculated as the product of frequency and wavelength. In this scenario, the speed of transverse waves on the rope is given as 62.0 m/s, which is crucial for determining the wavelength and frequency of the fundamental mode of vibration.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system vibrates. For a rope fixed at both ends, the fundamental frequency corresponds to the simplest standing wave pattern, with one half-wavelength fitting between the supports. This concept helps in calculating the frequency using the given wave speed and rope length.

Wavelength

Wavelength is the distance between consecutive points of a wave in phase, such as crest to crest. In the fundamental mode of a rope fixed at both ends, the wavelength is twice the length of the rope. Understanding this relationship allows us to calculate the wavelength using the rope's length and the wave speed.